Abstract

The Ulam-Hyers stability of functional equations is widely studied from various points of view by many authors. The present paper is concerned with local stability of the four Cauchy equations restricted on a bounded domain. These results can be easily adapted to the corresponding Pexiderized equations.

1. Introduction

After this introduction, in Section 2 the local stability of the additive equation
and, as a consequence, of the logarithmic equation
both restricted on a bounded domain in is studied.

It is well known that the problem of stability was posed, for the additive equation, by S. Ulam and was solved by Hyers [1] in 1941, with reference to the equation valid on the whole space. Afterwards, stability was widely studied by many authors, from various points of view, considering further equations on the whole space or putting them in very general settings (see, for instance, [2, 3]).

As for the “local” stability of equations on a restricted domain, first results can be found in [4, 5] (see also [6]) and they concern substantially the set of functions , satisfying the condition of -additivity
restricted either on the triangular domain
for some given , or the unbounded domain
for some given .

In the present paper (Section 2) the bounded restricted domain of inequality (3) will be assumed to be the triangle
for some given and .

It has to be remarked that in the classical paper [1] by Hyers as well as in case of restricted domains studied in [4, 5], the solutions of the “equation” correlated to the inequality (3) on the given domain are either additive functions on the whole space (in cases of the general result by Hyers and of the domain defined in (5)) or the restrictions to of functions additive on (in case of a domain like defined in (4)).

On the contrary, when a restricted domain like that in (6) is assumed, the local solution of the corresponding additive equation, restricted to the same set, is different from the restriction on the domain of of some function, which is additive in the whole space (see [7]).

Therefore, in order to adhere to the sense of Ulam’s question in case of a restricted domain like too, the locally -additive function has to be compared with the local solution of the corresponding exact equation restricted to the same set .

In this frame, in Section 2, first the local stability of the additive Cauchy equation restricted to will be proved (Theorem 1); then, as a consequence of this result, the local stability of the logarithmic Cauchy equation (2) will be proved (Theorem 6).

These results can be easily extended to the Pexiderized forms of the same equations.

Notice that the problem of the “local” stability for the remaining two Cauchy equations,
restricted to bounded domains, requires a suitable slightly different approach because of the peculiar properties of the local solutions of such equations when they are restricted on bounded domains (see [8, 9]).

This problem will be the object of Section 3 of the present paper, where results of local stability of (7) and (8) will be proved (Theorems 15 and 16, resp.).

2. About the Additive and the Logarithmic Cauchy’s Equations

2.1. A Result on Local Stability of the Additive Equation

In the set of functions of a real variable with values in a normed space , let us consider the inequality
for some and , defined in (6), for given and .

As usual, the projections of will be denoted by

For as defined in (6), the projections are , , , and the domain of functions satisfying (9) is .

We purpose to check whether each satisfying (9) for is uniformly approached on by some satisfying the additive equation restricted to , namely, by a function of the following form (see [7]):
where is additive on the whole space and are constant.

A positive answer is given by the following.

Theorem 1. Let , being a Banach space, satisfy (9) for some and every , defined in (6), for given and ; .Then there exists (at least) a function , additive on , such that the function defined by
has both the following properties:(i)is a (local) solution of the additive equation restricted on ;(ii) approaches uniformly on , and
holds for every .

Lemma 2. Let satisfy the condition (9) restricted to the set defined in (6). Then the functions , , defined by
for satisfy both the following inequalities:

Proof of Lemma 2. Let us prove (15) firstly for , . For the points and belong to ; hence, from (9) both the following inequalities hold
whence,
namely, (15) for , .Similarly we prove (15), for , , assuming the pairs and with , from the formulas
and for , assuming the pairs , , and .In order to prove (16) let us assume .For , from (15) and (9) we get
since and , it follows from (9) that
with , for , whence (16) for .Similarly, for , and interchanged.As for with , from (9)
where , , whence .Lemma 2 is proved.

Lemma 3. If , being a Banach space, satisfies (9) on , then each of the functions , , defined in (14) for , is uniformly approached on by the restriction of a function additive on .The following inequalities hold:

Proof of Lemma 3. Since each function , , is -additive on (from Lemma 2), formula (23) follows immediately from a known result [4, Lemma 2]:Let , a Banach space, be -additive on = .Then there exists at least one additive function such that
Moreover, for every and , formula (24) follows from (15) and (23); in fact
This means that the restriction to of each additive , , approaches uniformly on each function , , .Lemma 3 is proved.

Proof of Theorem 1. According to (24) in Lemma 3, each function , , defined in (14) for , namely,
is uniformly approached on by each of the additive functions , .Let us define , , as follows:
Such functions , , satisfy obviously the additive equation restricted to .Moreover, thanks to Lemmas 2 and 3, each function approaches uniformly on as in formula (13); in fact, for arbitrary ,
similarly for , .On the projection , where and , we get from Lemma 3 and formula (9)
Therefore, each function , , satisfies (13), and Theorem 1 is proved.

Remark 4. The foregoing study was developed as though the projections , , and were pairwise disjoint.If two of them overlap, for instance is nonempty, in every common point the values given by the different parts of formulas of approximating function have to be the same.More in particular, if the set is connected, in (28) the equations
hold, whence . In this case, the locally -additive is uniformly approached on the whole by the restriction to of a function , additive on (, either =1, or =2, or =3).

2.2. A Result on Local Stability of the Logarithmic Equation

On the ground of the results in Section 2.1 it is easy to prove the local stability of the logarithmic Cauchy equation (2) restricted to the bounded domain
for given and .

The projections , , of are given by

Since the local stability of (2) depends on the comparison of every satisfying
for with some solution of the corresponding equation (2) restricted to , let us premise (Lemma 5) the local solution of (2).

Lemma 5. Let be a real linear space; if satisfies (2) on the bounded domain defined in (32), then there exists a function , additive on , such that

Proof of Lemma 5. By the usual substitutions
the domain is transformed into a set like the one defined in (6). Let us consider
Equation (2) is transformed into
Therefore, using Theorem 1, we obtain (35) with additive .

The local stability of the logarithmic equation (2) is stated by the following.

Theorem 6. Let be a Banach space; if satisfies
for some and every , defined in (32), for given and , then there exists (at least) a function , additive on , such that the function defined by
satisfies both the following properties:(i) is a local solution of the logarithmic equation on the restricted domain ;(ii) approaches uniformly on , and
holds for every .

Proof of Theorem 6. The usual substitutions , (like in proof of Lemma 5) transform the inequality (39) restricted to the set into
restricted to the set , defined in (6).Now, we can follow the same line of proof as in Section 2.1 by defining the functions , , related to ; namely,
then there exist functions , , additive on , such that each of the functions , , is a local solution of the equation restricted to , and
holds for .Now let us come back to , by the substitutions which transformed (39) into (42), beginning by the transformation of functions defined in (44); on (from )
similarly, for on (from ) and for on (from ).By the definition
formula (44) changes intoObviously each , , satisfies the logarithmic equation restricted to .In order to prove the approximation stated in (41), let us begin by the projection : for ; then , , and ; hence, from (44)
and from (45)
Similarly for on and on .Therefore, (41) is true with or or , and Theorem 6 is proved.

Remark 7. Remarks about the consequence of a possible overlapping of the projections of the given restricted domain, like those in Remark 4, could be repeated here.

2.3. About the Pexiderized Forms of the Foregoing Equations

Stability results for the Pexiderized forms of the additive and the logarithmic equations, namely,
can be easily stated on the ground of the foregoing Theorems 1 and 6.

In fact, when the inequality
is satisfied for every , the statement of Theorem 1 can be easily adapted to the condition (52), because the functions , and play a role like that of the restrictions of to the projections , , in case of a unique function .

Remark 8. In case of a Pexiderized equation on restricted domain, overlapping of the projections of the given bounded domain obviously produces no changes in the result.

3. About the Remaining Two Cauchy Equations (7) and (8) on a Bounded Restricted Domain

3.1. Preliminaries

As for (7) the restricted domain is assumed to be defined in (6); the domain of (8) is defined in (32) for fixed real , and .

Let us premise the local solutions of the above equations (see papers [8, 9] and [10], resp.).

Lemma 9. Let satisfy (7) restricted to defined in (6).If and only if there exists some , such that and , the following properties (P1), (P2), (P3) hold: (P1) for every (hence for every ); (P2) is constant on each projection (not necessarily the same in different projections); (P3) is given on by the following formulas: (i)if , , (ii)if , , (iii)if , ,where is additive on ; , are constant.

Remark 10. Notice that restricted to each of the projections , , is the restriction of a solution of the equation
valid on the whole , for suitable . Since this equation can be written as
we get , for some additive , whence formulas in (P3) of Lemma 9 for
respectively, in , , .

Lemma 11 (see [10]). The general nowhere vanishing solution of (8) restricted to the set defined in (32) is given by the following formulas:
where is additive on .

Remark 12. As in Remark 10, we can see that the local solution of (8), restricted to each of the projections , , , is the restriction of a solution of a more general equation
for suitable . From , it follows that , with

3.2. How the Question of Local Stability of (7) or (8) Has to Be Properly Formulated?

The foregoing Remarks 10 and 12, which point out a connection of the restricted equation under consideration with more general equations, namely,
suggest the following forms of perturbation of such equations:
for and some fixed .

Moreover, it is known (see [8, 9]) that the local solutions of the restricted equations (7) or (8), which vanish somewhere, are expressed by formulas containing arbitrary functions; therefore, the problem of the local stability seems to be significant in the set of nowhere vanishing functions .

In this frame, the perturbed forms and can be written equivalently as

The stability results which follow are framed in this context.

3.3. A Sign Property concerning the Perturbed Forms of the Exponential Equation and the Power Equation

Here, we will be concerned with the condition , defined in (6) for some fixed , in the set of functions , such that for every .

Let us premise a remark about signs of nowhere vanishing functions satisfying on . From Lemma 9, Property (P2), it is known that every nowhere vanishing solution of the exponential Cauchy equation restricted to keeps a constant sign in each of the projections , , of .

We will see that a similar property is true also for every solution of the restricted condition , which is rewritten here as follows:
From , assuming and with , we get
whence

Moreover, from for = , , = , = ; hence, has constant sign in .

As a consequence, from (62), (64) it follows that has constant signs also in and in (the signs of and of , resp.).

This proves the following.

Lemma 13. Every nowhere vanishing function satisfying in keeps constant sign in each of the projections , , of .

Similarly, we can prove a sign property concerning the perturbed form of the power equation.

Let us consider now the condition ; namely,
for some fixed , assuming for every .

The usual substitutions of variables allow us to use the foregoing results about the exponential equation. Putwhence , .

Therefore, from Lemma 13, it follows that has constant sign (=) in , whence has constant sign (=) in ; similarly for in , namely for in and for in , namely in .

Hence, the following result is proved.

Lemma 14. Every nowhere vanishing function satisfying (65) restricted to has constant signs in each of the projections , , of .

3.4. A Result of Local Stability for the Exponential Cauchy Equation

In the set of functions such that for every , let us consider the inequality , with for some fixed .

From
since
satisfies
namely for ,

On the ground of Theorem 1, there exists (at least) one additive function , such that the function defined by
is a local solution of the additive Cauchy equation restricted to , such that

Since , whence,
we get
substitution of by its explicit formulas gives the following:and similarly

By defining
it is easily proved that

Moreover, from (78),
This means that the values of in are “near” (in dependence on ) the values of a local solution of the corresponding equation restricted to the same domain and give the following theorem of local stability.

Theorem 15. If the function , is nowhere vanishing in its domain and satisfies , for some given and every defined in (6) for given and , then there exists (at least) an additive function such that the function
has both the properties:(i) is a nowhere vanishing local solution of the exponential Cauchy equation restricted to ;(ii)the values are near the values on ; more exactly

3.5. A Result on Local Stability of the Power Cauchy Equation

In the set of nowhere vanishing functions , let us consider the inequality (65), defined in (32), for some given .

Hence, thanks to Theorem 15 (referred to instead of ), there exists (at least) one additive function such that the function defined by
satisfies the exponential equation restricted to and approaches on as follows:

Formula can be rewritten as
for .

From the definition of ,for then , ,similarly for then , ,and for : .

Hence, by defining as follows:
we get
namely,

This proves the following property of local stability of the “power” Cauchy equation.

Theorem 16. Let the nowhere vanishing function satisfy the condition (65) for some given and every defined in (32), for given and ; ; then there exists (at least) an additive function such that the function defined by
has both the following properties:(i) is a local solution of the Cauchy equation restricted to ;(ii)the values of are near the values in ; more exactly

3.6. Remark about the Pexiderized Forms of the Foregoing Equations

According to the remarks at the end of Section 2, the stability results given by Theorems 15 and 16 can be easily adapted to the Pexiderized forms of the corresponding equations, namely, to
for nowhere vanishing functions